Alice Quillen University of Rochester

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Origin Scenarios for Multiple Planet

Systems

June 2013

In collaboration with• Alex Moore• Imran Hasan• Eva Bodman• Richard Edgar

Kepler ObservatorySearch for Planetary Transits

in Light-curves(Carter et al. 2012)

Kepler 36b Kepler 36c

Multiple planet systems

The Kepler Multiple planet systems• Lower planet masses than Doppler (radial velocity discovered)

planets• closely packed, short periods, compact systems• nearly circular orbits• low inclinations • Statistically significant number of planet pairs near or in

resonance Kepler planet candidate pairs (Fabrycky et al. astroph 2012)

period ratio

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Orbital resonance

The ratio of orbital periods of two bodies are nearly equal to a ratio of small integers

using mean motions (angular rotation rates)

integrating to give a resonant angle

Three unique and very different multiple planet systems

• Kepler 36 – two transiting super-Earth planets in nearby orbits, near the

7:6 resonance and with extreme density contrast around a solar mass subgiant

• HR 8799 (discovered via optical imaging)– 4 massive super-Jovian planets, with a debris disk in a young

system around an A star, 3 planets in a chain of mean motion resonances 4:2:1

• KOI 730 (Kepler candidate system)– 4 transiting super-Earth planets in a chain of mean motion

resonances around a Solar type star, 8:6:4:3 commensurability

What do the resonant systems tell us about planetary system formation and

evolution?• Resonant systems can be delicate constraints on

asteroid/planetesimal belts that can nudge planets out of resonance

• Resonances are narrow. Migration of planets allows capture into resonance constraints on migration processes – Pioneering work on this connection by Man-Hoi Lee in

2002

Transit Timing VariationsFigure: Agol et al. 2004

• Length of a transit gives a measurement for the radius of a planet, not its mass. • Transit timing variations allow measurement of planet masses! • Compact or/and resonant transiting systems give measurable transit timing

variations. Planetary sized masses can be confirmed.• Both planetary masses and radii are measured in the Kepler 36 system

Shift in location of center of mass of internal system causes a change in the time of the transit of outer planet

star + two planets

Transit timing variations in the Kepler 36 system

Fits to the transit timing make it possible to measure the masses of both planets

Carter et al. 2012

Kep 36b transits Kep 36c transits

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Mass Radius relation of Kepler planets

Other exoplanetsblue, Kepler-11 pink, Kepler-18b gray, Kepler-20 b and c brown, GJ 1214bviolet, CoRoT-7b green, Kepler-10b orange, 55 Cnc e

Carter et al. 2012

Kepler 36c outer planet fluffball

Kepler 36b inner planet solid rock+iron!

Kepler planets have a wide distribution of densities and so compositions!

Kepler 36 system

Two planets, near the 7:6 resonance

Large density contrast

Carter et al. (2012)

measured via astro-seismology

inner planet

outer planet

Quantities in the Kepler 36 system

• Ratio of orbital periods is 1.1733 (7/6=1.1667) • Distance between planets at conjunction is only

4.8 Hill radii! (Chaotic dynamics: Deck et al. 2012)• Planet sizes are large compared to volume: Integrations must

check for collisions• Circular velocity is ~90 km/s

Planet b Planet c

Planet mass/Stellar mass 1.15x10-5 2.09x10-5

Orbital velocity/Escape velocity 4.8 5.3

Semi-major axis /Hill radius 63.9 52.3

Hill radius/Planet radius 29.0 16.0

Semi-major axis/Planet radius 1852 838

Planetary Migration Scenarios

• A planet embedded in a gas disk drives spiral density waves

• Damps the planet’s eccentricity

• The planet usually moves inwards

• facilitates convergent migration and resonance capture

Phil Armitage

planet

Migration via Scattering Planetesimals

• A planet can migrate as it ejects and scatters planetesimals

• Facilitates divergent migration Pulling planets out of resonance or resonance crossing

Kirsh et al. 2009

semi-major axis in AU

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Stochastic migration• Planet receives little

random kicks • Due to density

variations from turbulence in the gas disk (e.g., Ketchum et al. 2011)

• Due to scattering with planetesimals (e.g., previously explored for Neptune by R. Murray-Clay and J. Hahn)

Jake Simon

Mean motion ResonancesCan be modeled with a pendulum-like Hamiltonianθ Resonant angle. Two types of motion, librating/oscillating in or out of resonance

expand Kepler Hamiltonian

due to two-planet interactions

Level curves showing orbits

This model gives:resonant width, strength, libration frequency, adiabatic limit, eccentricity variation in resonance, probability of capture

Dimensional Analysis on the Pendulum

• H units cm2 s-2

• Action variable p cm2 s-1 (H=Iω) and ω with 1/s

• a cm-2

• b s-1

• Drift rate db/dt s-2

• ε cm2 s-2

Ignoring the distance from resonance we only have two parameters, a,εOnly one way to combine to get momentumOnly one way to combine to get time Distance to resonance

Sizescales on the Pendulum

• Libration timescale ~ • Momentum variation in resonance• Distance to resonance• Adiabatic limit• Critical eccentricity set from momentum scale

Dimensional analysis on the Andoyer Hamiltonian• We only have two important parameters if we ignore

distance to resonance

• a dimension cm-2

• ε dimension cm2-k s-2-k/2

• Only one way to form a timescale and one way to make a momentum sizescale.

• The square of the timescale will tell us if we are in the adiabatic limit

• The momentum sizescale will tell us if we are near the resonance (and set critical eccentricity ensuring capture in adiabatic limit)

First order Mean motion resonances

Two regimes: High eccentricity: We model the system as if it were a pendulum with

Low eccentricity :Use dimensional analysis for Andoyer Hamiltonian in the low eccentricity limit

dividing line dependent on dimensional eccentricity estimate

Before resonance capture we work with the low eccentricity dimensionsAfter resonance capture we work with the pendulum models.

Can the Kepler 36 system be formed with convergent migration?

• Two planet + central star N-body integrations

• Outer planet migratesdamping is forced by adding a drag term in the integration

• Eccentricity dampingforced circularization using a drag term that depends on the difference in velocity from a circular orbit

4:3 resonance

apsidal angle = 0 in resonance (see Zhou & Sun 2003, Beauge & Michtchenko, many papers)

semi-major axes with peri and apoapses

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Drift rates and Resonant strengths

• If migration is too fast, resonance capture does not occur

• Closer resonances are stronger. Only adiabatic (slow) drifts allow resonance capture.

• Can we adjust the drift rate so that 4:3, 5:4, 6:5 resonances are bypassed but capture into the 7:6 is allowed?

• Yes: but it is a fine tuning problem. The difference between critical drift rates is only about 20%

Eccentricities and Capture• High eccentricity systems

are less likely to capture • Can we adjust the

eccentricities so that resonance capture in 4:3, 5:4, 6:5 resonances is unlikely but 7:6 possible?

• No. Critical eccentricities differ by only a few percent.

capture into 3:2 prevented by eccentricities

Secular oscillations and resonance crossings make it impossible to adjust eccentricities well enough

resonances are bypassed because of eccentricities

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secular oscillations

eccentricity jump due to 7:5 resonance crossing

Stochastic migration• Does stochastic migration

allow 4:3, 5:4, and 6:5 resonances to be bypassed, allowing capture into 7:6 resonance?

• Yes, sometimes (also see work by Pardekooper and Rein 2013)

• Random variations in semi-major axes can sometimes prevent resonance capture in 4:3, 5:4, 6:5 resonances

resonances bypassed

capture into 7:6!

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Rein(2013) accounts for distribution of period ratios of planet pairs using a stochastic migration model

Problems with Stochastic migration• Stochastic

perturbations continue after resonance capture

• System escapes resonance causing a collision between the planets

planets collide!

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Problems with Stochastic migration• If a gas disk causes both migration and stochastic forcing,

then planets will not remain in resonance• Timescale for escape can be estimated using a diffusive

argument at equilibrium eccentricity after resonance capture• Timescale for migration is similar to timescale for resonance

escape Disk must be depleted soon after resonance capture to account for a system in the 7:6 resonance --- yet another fine tuning problem

• Density difference in planets not explained

Collisions are inevitableKepler Planets are close to their starConsider Planet Mercury, closest planet to the Sun• Mercury has a high mean

density of 5.43 g cm-3 Why?– Fractionation at formation

(heavy condensates) – afterwards slowly,

(evaporation) – afterward quickly

(collision)• See review by Benz 2007

MESSENGER image

Giant Impact Origin of Mercury

Grazing collision stripped the mantle, leaving behind a dense core that is now the planet Mercury (Benz et al. 2008)

Figures by Asphaug (2010)

direct collision grazing collision

Geometry of collisions

hit and run,mantle stripping

Asphaug(2010)

envelope stripping

impact angle

slow collisions fast collisions

Planetary embryos in a disk edge

• ``Planet trap’’ + transition disk setting (e.g., Moeckel & Armitage 2012, Morbidelli et al. 2008, Liu et al. 2011)

• We run integrations with two planets + 7 embryos (twice the mass of Mars)

• no applied stochastic forcing onto planets, instead embryos cause perturbations

• The outermost planet and embryos external to the disk edge are allowed to migrate

Embryos can lie in the disk here!

Zhang & Zhou 2010

Integration ends with two planets in the 7:6 resonance and in a stable configuration

Collisions with inner planet. Potentially stripping the planet in place

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encounter with embryos nudge system out of 3:2 resonance

embryos migrate inwards

two planets

Integrations of two planets and Mars mass embryos

encounters with embryos nudge system out of 3:2, 5:4 resonances

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another integration Inner and outer planet swap locationsOuter planet that had experienced more collisions becomes innermost planet

Integration ends with two planets in the 6:5 resonance and in a stable configuration

Integration ends with two planets in the 4:3 resonance and an embryo in a 3:2 with the outer planet

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Final state can be a resonant chain like KOI 730

another integration

If a misaligned planet existed in the Kepler 36 system it would not have been seen in transit

Diversity of Simulation Outcomes• Pairs of planets in high j resonances such as 6:5 and 7:6.

Appear stable at end of simulation• Pairs of planets in lower j resonances such as 4:3• Resonant chains• Collisions between planets• Embryo passed interior to two planets and left thereComments• Collisions affect planetary inclinations -- transiting objects are

sensitive to this• A different kind of fine tuning: Numbers and masses of

embryos. Outcome sensitive to collisions!

Properties of collisions between embryos and planets

vimpact/vcircular

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impacts on inner planet especially likely to cause erosion

Accretion may still occur

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Impact angle (degrees)

Impacts are grazingImpacts are normal

High velocity, grazing impacts are present in the simulation suggesting that collisions could strip the envelope of a planet

Kepler 36 and wide range of Kepler planet densities

Both planet migration and collisions are perhaps happening during late stages of planet formation, and just prior to disk depletion …

Resonant Chains• Prior to the discovery of GL876 and HR8799, the only known

multiple object system in a chain of mean motion resonances was Io/Europa/Ganymede

• Each pair of bodies is in a two body mean motion resonance• Integer ratios between mean motions of each pair of bodies• Convergent migration model via tidal forces for Galilean

satellites resonance capture (and for a more complicated story see Cathy)

Resonant Chains

• Systems in chains of resonances drifted there by convergent migration through interaction with a gaseous disk (e.g. Wang et al. 2012)

• Scattering with planetesimals usually causes planet orbits to diverge and so leave resonance

What constraints can resonant chain systems HR8799 and KOI730 give us on their post gaseous disk depletion evolution?

KOI 730 systemresonant chain

• Planet masses estimated from transit depths

• Period ratios obey a commensurability8:6:4:3

• Outer and inner pair in 4:3 resonance

• Middle pair in 3:2

Discovered in initial tally of multiple planet Kepler candidates (Lissauer et al. 2011)

KOI-730 system• Suppose after formation the KOI730 system hosts a debris

disks of planetesimals. Could planet-orbit-crossing planetesimals (comets) pull the system out of resonance?

• How are planetary inclinations affected? To see 4 planets in transit, mutual inclinations must lie within a degree – Find resonant initial conditions– Run N-body integrations (GPU accelerated) with planetesimals

that are initially located in a disk exterior to the planets– We ran different simulations with different planetesimal disk

masses

Finding Initial ConditionsForced migration Capture into 8:6:4:3

Lots of eccentricity damping required to keep this system stableFine tuning in initial conditions and migration rates required

Capture of one pair often caused another pair to jump out of resonance

An integration that succeeded in giving the proper period ratios

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Initial conditions for our N-body integration taken here!

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KOI 730 Simulations

Simulation Mass of planetesimal disk

Orbit crossing Mass in Earth Masses

N Neptune Mass 16.6

N5 1/5 Neptune Mass 1.7

E Earth Mass 0.46

E3 1/3 Earth Mass 0.12

M Mars Mass 0.04

Z No planetesimals 0

Mass in planetesimals that crossed the planets’ orbits was measured

Changes in period ratiosmassive planetesimal disk, planets out of resonance

less planetesimal mass, system still in resonance

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Resonances are crossed, causing of increases in eccentricities and inclinations

inclinations do not damp to zero as would be expected from dynamical friction

massive planetesimal disk

less planetesimal mass

Trends seen in the simulations• A Mars mass or orbit crossing planetesimals pulls the

system out of resonance. • This can be ruled out for KOI-730! Less than a Mars mass

in planetesimals could have crossed the orbits of the KOI-730 planets

• An Earth mass of orbit crossing planetesimals, puts system just outside resonance, by an amount similar to the peak seen in a histogram of Kepler system period ratios.

• Correlation between orbit crossing mass and inclinations to look for with Kepler observations

HR8799 system

6 — 1000 AU

• HR 8799, A star, young! • Hosts a debris disk• 4 massive planets• Discovered via optical imaging

Marois et al. 2011

evidence of debris

HR8799 simulations

• Using orbital elements based on observed positions of planets

• Different mass planetesimal disks• Start with an unstable planetary configuration. Can

the planetesimal disk can stabilize the system via eccentricity damping? No: Too much disk mass is required to make this possible

• Start with a stable planetary configuration. Can the planetesimals pull it out of resonance, causing instability?

Interaction between the HR8799 resonant chain and an external debris disk

A Neptune mass debris disk can substantially reduce the lifetime of the system.

Lifetimes with a Neptune Mass debris disk

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lifetime without a debris disk

Moore & Quillen 2012

HR 8799 planetary system stability

Gozdziewski & Migaszewski (2009)

stable unstable Maximally stable configurations have planets c,d,e in a 1:2:4 resonant configuration (Gozdziewski & Migaszewski 2009, Fabrycky & Murray-Clay 2010, Marois et al. 2011)

• Lifetime of resonant configuration is short (order 107 years)

• Planets likely will be ejected from the system (perhaps soon!)

• Zone of stability is very small

HR 8799 planetary system stability causes

Gozdziewski & Migaszewski (2009)

stable unstable

The system is currently observed to be at the boundary of stability. It might be at this boundary because planetesimal mass has pulled it away from the bottom of the resonance

Even though the planets are massive, the stable region is very small so a very small amount of debris affects stability

Summary: Kepler 36 Origins• Stochastic migration scenarios to account for Kepler 36’s

origin require fine tuning so that planets can bypass 4:3, 5:4, 6:5 resonances and capture into the 7:6 resonance. Stochastic forcing would pull the system out of resonance unless the gas disk is depleted soon after capture

• Encounters with planetary embryos can remove two planets from outer resonances allowing them to end up in adjacent orbits like Kepler 36b,c. Impacts with embryos can have high enough velocity and impact angles that the mantle of a planet could be stripped, leaving behind a high density core. This scenario can account for both the proximity of the Kepler 36 planets and their high density contrast

Summary:Constraints on planetesimal disks

• KOI-730: Less than a Mars mass of planetesimals could have crossed the orbits of planets, otherwise the 4 planet system would be pulled out of resonance, and planet inclinations increased past those observed A compact Kepler system never interacted with debris after the disk depleted (no solar system shake up)

• HR8799: Is near instability, a 1/10th of a planet mass can pull the system out of resonance causing it to fall apart Its debris disk (observed) could be responsible for system’s current location at the edge of stability